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Chapter 7: Problem 103

Determine the domain of each function described. $$ f(t)=7+\sqrt[8]{t^{8}} $$

Short Answer

Expert verified
The domain is \( \mathbb{R} \).

Step by step solution

01

Understand the function given

The function provided is \( f(t) = 7 + \sqrt[8]{t^{8}} \). It combines a constant with an 8th root expression.
02

Simplify the function

Simplify \( \sqrt[8]{t^{8}} \). We know that the root and the exponent cancel each other out under certain conditions. So, \( \sqrt[8]{t^{8}} = |t| \). This is because taking the 8th root of any number raised to the power of 8 gives the absolute value of that number.
03

Analyze the simplified function

After simplification, we have \( f(t) = 7 + |t| \). The absolute value function \( |t| \) and the constant 7 do not restrict the domain of the function.
04

Determine the domain

Since the function \( f(t) = 7 + |t| \) does not have any restrictions such as division by zero or taking the square root of a negative number, the domain is all real numbers.
05

State the final domain

Thus, the domain of the function \( f(t) = 7 + \sqrt[8]{t^{8}} \) is \( \mathbb{R} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always non-negative. For any real number \(t\), the absolute value is denoted as \(|t|\). Here's why it's important in our function:
  • When simplifying \(\sqrt[8]{t^{8}}\), we get \(|t|\). This is because raising a number to the eighth power makes it non-negative, and the eighth root of a non-negative number is the absolute value of the original number.
  • Thus, \(\sqrt[8]{t^{8}}\) ensures we're always working with non-negative values, even if \(t\) itself is negative.


Understanding absolute values helps us see why \(\sqrt[8]{t^{8}} = |t|\) plays no role in restricting the function's domain.
root and exponent simplification
Root and exponent simplification is a key algebraic skill. It allows us to reduce complex expressions into something simpler and easier to work with.
  • In the function \(f(t) = 7 + \sqrt[8]{t^{8}}\), the root and exponent cancel each other out due to the properties of exponents and radicals.
  • Specifically, \(\sqrt[8]{t^{8}}\) simplifies to \(|t|\). Why? Because the eighth root of \(t^{8}\) is the same as asking which number, when raised to the eighth power, gives us \(t^{8}\). That number is \(|t|\).

By using these simplification techniques, we turn \(f(t) = 7 + \sqrt[8]{t^{8}}\) into \(f(t) = 7 + |t|\), making it clearer and easier to handle.
real numbers domain
The domain of a function is the set of all possible input values (t) that the function can handle without running into problems like division by zero or taking the square root of a negative number.
  • For \(f(t) = 7 + |t|\), we need to decide if there are any \(t\) values that cause issues. Fortunately, there aren't any.
  • The absolute value \(|t|\) is defined for all real numbers. Adding 7 doesn't introduce any new restrictions.

This means the domain of \(f(t) = 7 + \sqrt[8]{t^{8}}\) (which simplifies to \(f(t) = 7 + |t|\)) is all real numbers. We write this as \(\mathbb{R}\), indicating that any real number is a valid input for \(t\).

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Most popular questions from this chapter

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